Existence of Bade functionals for complete Boolean algebras of projections in Fréchet spaces

A classical result of W. Bade states that if is any complete Boolean algebra of projections in an arbitrary Banach space then, for every there exists an element (called a Bade functional for with respect to in the dual space , with the following two properties: (i) is non-negative on and, (ii) whenever satisfies It is shown that a Fréchet space has this property if and only if it does not contain an isomorphic copy of the sequence space

     

Grothendieck operators on tensor products

We prove that for Banach spaces and operators , the tensor product is a Grothendieck operator, provided is a Grothendieck operator and is compact.

     

Semi-free actions of zero-dimensional compact groups on Menger compacta

Let be the -dimensional universal Menger compactum, a -set in and a metrizable zero-dimensional compact group with the unit. It is proved that there exists a semi-free -action on such that is the fixed point set of every . As a corollary, it follows that each compactum with can be embedded in as the fixed point set of some semi-free -action on .

     

Lifting of generating subgroups

Let be an epimorphism of finite groups. Suppose that is generated by its subgroups and that is generated by its subgroups . Furthermore, suppose that and are conjugate, . We prove that there exist such that generate and , .

     

On the rational cuspidal subgroup and the rational torsion points of

For two distinct prime numbers , , we compute the rational cuspidal subgroup of and determine the -primary part of the rational torsion subgroup of the old subvariety of for most primes . Some results of Berkovic on the nontriviality of the Mordell-Weil group of some Eisenstein factors of are also refined.

     

A commutativity theorem for semibounded operators in hilbert space

Let and be semibounded (bounded from below) operators in a Hilbert space and a dense linear manifold contained in the domains of , , , and , and such that for all in . It is shown that if the restriction of to is essentially self-adjoint, then and are essentially self-adjoint and and commute, i.e. their spectral projections permute.

     

On genera of smooth curves in higher dimensional varieties

We prove that for any smooth projective variety of dimension , there exists an integer , such that for any integer , there exists a smooth curve in with .

     

Rigidity of compact manifolds with boundary and nonnegative Ricci curvature

Let be an ()-dimensional compact Riemannian manifold with nonnegative Ricci curvature and nonempty boundary . Assume that the principal curvatures of are bounded from below by a positive constant . In this paper, we prove that the first nonzero eigenvalue of the Laplacian of acting on functions on satisfies with equality holding if and only if is isometric to an -dimensional Euclidean ball of radius . Some related rigidity theorems for are also proved.

     

Commutativity criteria for Banach -algebras

Let be a Banach -algebra with an identity. Necessary and sufficient conditions are given for to be commutative modulo its -radical and for to be commutative if has a faithful -representation as operators on a Hilbert space.

     

A conformal differential invariant and the conformal rigidity of hypersurfaces

For a hypersurface of a conformal space, we introduce a conformal differential invariant , where and are the first and the second fundamental forms of connected by the apolarity condition. This invariant is called the conformal quadratic element of . The solution of the problem of conformal rigidity is presented in the framework of conformal differential geometry and connected with the conformal quadratic element of . The main theorem states:

Let , and let and be two nonisotropic hypersurfaces without umbilical points in a conformal space or a pseudoconformal space of signature . Suppose that there is a one-to-one correspondence between points of these hypersurfaces, and in the corresponding points of and the following condition holds: where is a mapping induced by the correspondence . Then the hypersurfaces and are conformally equivalent.

     

Poincaré flows

We study flows on a compact metric space with the property that corresponding to every non-zero element of there is either a cross section associated with or one associated with . We obtain necessary and sufficient conditions for this to hold; on the -dimensional torus these conditions take a classical form.

     

Collapsing successors of singulars

Let be a singular cardinal in , and let be a model such that for some -cardinal with . We apply Shelah's pcf theory to study this situation, and prove the following results. 1) is not a -c.c generic extension of . 2) There is no ``good scale for ' in , so in particular weak forms of square must fail at . 3) If then and also . 4) If then .

     

The Jung Theorem in metric spaces of curvature bounded above

The classical Jung Theorem states in essence that the diameter of a compact set in satisfies where is the circumradius of . The theorem was extended recently to the hyperbolic and the spherical -spaces. Here, the estimate above is extended to a class of metric spaces of curvature introduced by A. D. Alexandrov. The class includes the Riemannian spaces. The extended estimate is of the form where is a positive integer suitably defined for the set and its circumcenter. It can be that is not unique or does not exist. In the latter case, no estimate is derived. In case of a Riemannian -dimensional space, an integer always exists and satisfies . Then . In case of , one has .

     

A formula with nonnegative terms for the degree of the dual variety of a homogeneous space

Let be a reductive group and a parabolic subgroup. For every -regular dominant weight let denote the variety embedded in the projective space by the embedding corresponding to the ample line bundle . Writing , we prove that the degree of the dual variety to is a polynomial with nonnegative coefficients in . In the case of homogeneous spaces we find an expression for the constant term of this polynomial.

     

Invariants of Skew Derivations

If is an automorphism and is a -derivation of a ring , then the subring of invariants is the set The main result of this paper is Theorem. Let be a -derivation of an algebra over a commutative ring such that

for all , where and .

(i)

If , then .

(ii)

If is a -stable left ideal of such that , then .

This theorem generalizes results on the invariants of automorphisms and derivations.

     

On the Ganea conjecture for manifolds

Using a result of Singhof, we prove that provided is a connected closed PL manifold with and is the -sphere, .

     

On reducibility of semigroups of compact quasinilpotent operators

The following generalization of Lomonosov's invariant subspace theorem is proved. Let be a multiplicative semigroup of compact operators on a Banach space such that for every finite subset of , where denotes the Rota-Strang spectral radius. Then is reducible.

This result implies that the following assertions are equivalent:

(A) For each infinite-dimensional complex Hilbert space , every semigroup of compact quasinilpotent operators on is reducible.

(B) For every complex Hilbert space , for every semigroup of compact quasinilpotent operators on , and for every finite subset of it holds that .

The question whether the assertion (A) is true was considered by Nordgren, Radjavi and Rosenthal in 1984, and it seems to be still open.

     

On subalgebras of Boolean interval algebras

We prove that the following three conditions are necessary and sufficient for a Boolean algebra to be embeddable into an interval algebra.

(i)

is generated by a subset such that for all .

(ii)

has a complemented subalgebra lattice, where complements can be chosen in a monotone way.

(iii)

is isomorphic to ClopX for a compact zero-dimensional topological semilattice such that for all .

     

On weighted inequalities for singular integrals

In this note we consider singular integrals associated to Calderón-Zygmund kernels. We prove that if the kernel is supported in then the one-sided condition, , is a sufficient condition for the singular integral to be bounded in , , or from into weak- if . This one-sided condition becomes also necessary when we require the uniform boundedness of the singular integrals associated to the dilations of a kernel which is not identically zero in . The two-sided version of this result is also obtained: Muckenhoupts condition is necessary for the uniform boundedness of the singular integrals associated to the dilations of a general Calderón-Zygmund kernel which is not the function zero either in or in .

     

On invertibility in non-selfadjoint operator algebras

Let be a complete commutative subspace lattice on a Hilbert space. When is purely atomic, we give a necessary and sufficient condition for for every in , where and denote the spectrum of in and respectively. In addition, we discuss the properties of the spectra and the invertibility conditions for operators in .